Techniques for communicating over single-or multiple-antenna channels having both temporal and spectral fluctuations

ABSTRACT

This inventions provides techniques for estimating both temporal and spectral channel fluctuations with the duration of a data symbol. Certain pulse shaping functions are Discrete Prolate Spheroidal Sequences (DPSSs) and are used primarily because of their relatively limited Inter-Symbol Interference (ISI) properties. During reception, these properties allow one or more parameters of a joint time-frequency channel model to be more easily determined. Once the one or more parameters are determined, they can be applied to received symbols to correct the temporal fluctuations, spectral fluctuations, or both of the channel over which a communication took place. The techniques may be adapted for the Multiple-In, Multiple-Out communication situation.

FIELD OF THE INVENTION

[0001] The present invention relates generally to communications overwireless channels, and more particularly, to techniques forcommunicating over single or multiple antenna channels having bothtemporal and spectral fluctuations.

BACKGROUND OF THE INVENTION

[0002] In a single antenna wireless communication, a single transmittercommunicates with a single receiver over a wireless channel. In amultiple antenna wireless communication, multiple transmitterscommunicate with one or more receivers over a wireless channel. Eachreceiver can determine multiple outputs corresponding to the multipletransmitters. This is commonly called Multiple-In, Multiple-Out (MIMO),and is a relatively recent addition to the wireless field.

[0003] Ideally, any transmitted signal appears at the receiver as areceived signal, where the transmitted and received signals areidentical except for a time delay caused by the time the transmittedsignal takes to traverse the channel. However, many wireless channelscause fluctuations in the received signal. For instance, a receiver maybe moving away from the transmitter, which will cause a temporalfluctuation. As another example, the transmitted signal may bounce offof multiple buildings on the way to the transmitter. These different andslightly delayed signals can cause destructive or constructiveinterference. This type of interference can cause spectral fluctuations.

[0004] There are techniques for compensating for these fluctuations. Forinstance, many techniques send “pilots,” which are known symbols, duringa communication. A receiver uses the pilots to attempt to correct foreither temporal or spectral fluctuations, but not both. However, pilotusage can get very high for the MIMO communication systems. Each pilottakes the space of data, so more pilots means less bandwidth for datatransmission. Moreover, each of these techniques to correct temporal orspectral fluctuations of wireless channels causes its own set ofproblems. A need therefore exists for techniques that correct fortemporal and spectral fluctuations and also allow for MIMO communicationsystems.

SUMMARY OF THE INVENTION

[0005] Generally, the present invention provides techniques forcommunicating over single- or multiple-antenna channels, where thechannels can vary both temporally and spectrally. Basically, certainpulse shaping functions are used to modulate input symbols duringtransmission. The pulse shaping functions are Discrete ProlateSpheroidal Sequences (DPSSs) and are used primarily because of theirrelatively limited Inter-Symbol Interference (ISI) properties. Duringreception, these properties allow one or more parameters of a jointtime-frequency channel model to be more easily determined. Once the oneor more parameters are determined, they can be applied to receivedsymbols to correct the temporal fluctuations, spectral fluctuations, orboth of the channel over which a communication took place.

[0006] In a first aspect of the invention, techniques are disclosed forpreparing and, if desired, transmitting data using Discrete ProlateSpheroidal Sequences (DPSSs). Symbols are modulated using the DPSSs andthen combined into a transmitter signal. The transmitter signal is thengenerally transmitted. In one embodiment, certain DPSSs are chosen totransmit data symbols. Additionally, others of the DPSSs are chosen tomodulate pilot symbols, which are known symbols. Furthermore, a “guardsymbol” may be modulated by a particular DPSS chosen for its orderbetween a group of pilot symbols and a group of data symbols. In anadditional embodiment, the DPSSs chosen to modulate the pilot symbolsare chosen to have the highest orders of all of the symbols used tomodulate the data symbols, the pilot symbols, and, if used, the guardsymbol. In another embodiment, for a predetermined time duration andbandwidth, the number of DPSSs is chosen such as to keep spill-over verylow (beneficially under five percent) while at the same time having aslarge a number of information-carrying symbols as is possible under thisenergy constraint.

[0007] In another embodiment, techniques are disclosed for decodingdata. One or more parameters of a joint time-frequency channel model aredetermined, and the one or more parameters are applied to demodulatedsymbols to determine decoded symbols. In one embodiment, certaindemodulated symbols correspond to pilot symbols. These certaindemodulated symbols are used to determine the parameters. This usuallyoccurs by using a system of equations, generally represented asmatrices. In one embodiment, the parameters determine the time slope,frequency slope, and amplitude of a plane used to estimate the temporaland spectral fluctuations of the channel. Once one or more parametersare known, the parameters can be applied to remaining demodulatedsymbols. Application of the parameters takes place through a system ofequations, generally represented through matrices.

[0008] The techniques of the first and second aspects may be extended tobe used in a MIMO system of a third aspect of the present invention,where multiple transmitters communicate with multiple receives overmultiple communication channels.

[0009] A more complete understanding of the present invention, as wellas further features and advantages of the present invention, will beobtained by reference to the following detailed description anddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0010]FIG. 1 is a chart illustrating how global time-frequency resourcesare divided into time-frequency blocks;

[0011]FIGS. 2A and 2B are charts of magnitude and phase, respectively,of an acoustic channel transfer function, used to illustrate a timetranslational invariant channel;

[0012]FIGS. 3A and 3B are charts of magnitude and phase, respectively,of two tones, used to illustrate a stochastic channel with “flatfading”;

[0013]FIG. 4A is a chart illustrating a possible actual time- andfrequency-variant transfer function of a physical channel;

[0014]FIG. 4B is a chart illustrating a prior art technique for modelinga frequency-varying transfer function;

[0015]FIG. 4C is a chart illustrating a prior art technique for modelinga time-varying transfer function;

[0016]FIG. 4D is a chart illustrating a technique of the presentinvention, which models a transfer function that varies in both time andfrequency, in accordance with a preferred embodiment of the invention;

[0017]FIGS. 5A and 5B are charts illustrating the time and frequency,respectively, operators, shown in the time basis;

[0018]FIGS. 6A through 6F are charts illustrating the first six DPSSs,used in aspects of the present invention;

[0019]FIG. 7 is a chart illustrating the energy concentration of theDPSSs within a bandwidth B, used to determine an appropriate amount ofDPSSs for transmitting data in accordance with a preferred embodiment ofthe invention;

[0020]FIGS. 8A and 8B are charts illustrating the time and frequency,respectively, operators, shown in the DPSS basis, and used to decodedemodulated symbols in accordance with a preferred embodiment of theinvention;

[0021]FIG. 9 is a block diagram showing a receiver and transmittercommunicating over a wireless channel in accordance with a preferredembodiment of the invention;

[0022]FIG. 10 is a diagram showing portions of a receiver andtransmitter that are communicating over a wireless channel in accordancewith a preferred embodiment of the invention;

[0023]FIG. 11 is a block diagram showing pilot placement on the K DPSSpulses in a single transmission burst, in accordance with a preferredembodiment of the invention;

[0024]FIG. 12 is a diagram showing portions of multiple receivers andmultiple transmitters that are communicating over a wireless channel inaccordance with a preferred embodiment of the invention; and

[0025]FIG. 13 is a block diagram showing pilot placement on the K DPSSspulses in multiple transmission burst, in accordance with a preferredembodiment of the invention.

DETAILED DESCRIPTION

[0026] Aspects of the present invention provide techniques forestimating the temporal and spectral fluctuations of a wireless channeland using the estimations to compensate for much of the signaldegradation damage caused by the fluctuations. Basically, the channel ismodeled through a joint time-frequency channel model, which is usuallyrepresented as containing a time-varying transfer function that relatestransmitted symbols to received symbols.

[0027] By using pilot symbols, the location and values of which areknown to both the receiver and the transmitter, parameters of the jointtime-frequency channel model are determined. The parameters determinedare the parameters of the time-varying transfer function. The parametersbasically define aspects of a plane used to model the time-varyingtransfer function of the channel. The parameters define the amplitude(where the center of the plane is located), and the slopes in time andfrequency. Once the parameters are determined, the model of the channelis fixed for a particular transmission block and the parameters areapplied to demodulated symbols to create decoded symbols. This processhas been shown to reduce errors caused by temporal and spectralfluctuations of the channel.

[0028] It should be noted that aspects of the present invention willcorrect for channel fluctuations, even if the channel varies onlytemporally or only spectrally. Additionally, the representations of the“plane” created with the model of the channel used herein are shown forpurposes of explanation only. Because the channel model usescomplex-valued parameters, the plane created by these parameters issimply a technique used to visualize aspects of the present invention.This visualization of planes is mathematically not exact in the sensethat the temporal evolution of a signal or system is linked to itsspectral content by the Fourier transform. It is therefore not possibleto specify both temporal and spectral content totally independently ofone another as could be suggested by the image of a plane totally flatin the direction of both time and frequency.

[0029] In technical terms, conventional techniques deal withtime-invariant channels within the data symbol. In this disclosure, a“data symbol” is the temporal span of the smallest unit of data that issent. It is not that conventional channel models do not allow for timevariation. Instead, it is that the variation does not occur during aparticular symbol. In conventional techniques, the channel is allowed tofluctuate symbol to symbol, in principle, and can be re-estimated foreach symbol, in principle.

[0030] By contrast, the present invention allows for a time variantchannel model within individual data symbols. Thus, the presentinvention proposes a channel model that cannot be represented as apurely frequency varying channel within an individual data symbol. Itshould be noted, however, that the channel model proposed herein mayhave a zero time slope coefficient, further described below, which meansthat, for one or more individual data symbols, the time-variance isdetermined as being zero.

[0031] A “time variant” channel is by definition a channel that is not“time invariant.” In other words, time variance is defined by negatingits opposite. A time invariant channel is one whose properties do notdepend on absolute time. Briefly, a linear channel is given by thefollowing input-output relation (which is described in greater detailbelow):${y(t)} = {\sum\limits_{t^{\prime}}^{\quad}{{h( {t,t^{\prime}} )}{{x( t^{\prime} )}.}}}$

[0032] For a time invariant channel, the transfer function h(t,t′) is afunction only of the difference time, t−t′. In other words,h(t,t′)=f(t−t′). A channel that cannot be written in this form is timevariant. A well known set of terminology that describes time invariantand time variant is “stationary” and “non-stationary,” respectively.See, for instance, Proakis, “Digital Communications,” McGraw-Hill, 64-64(3rd ed. 1995), the disclosure of which is hereby incorporated byreference.

[0033] The present invention proposes a more general form of transferfunctions, and suggests appropriate parameterizations. One specificproposal described herein is to express the transfer function in theSlepian basis and use parameters of the transfer function to correcttime and frequency variations. The techniques presented herein areapplicable to many components of a communication system, such astransmitters, receivers, modulation systems, and channel estimators.

[0034] For ease of reference, the disclosure is divided into sections:Section 1 gives information on why time-frequency channel estimation isused; Section 2 describes the first-order time frequency model ofaspects of the present invention; and Section 3 describes exemplaryapparatus used to carry out the invention and implementation issues.

[0035] 1 Why Time-Frequency Channel Estimation?

[0036] This section describes conventional techniques for channelestimation, describes the problems associated with these techniques, andintroduces techniques of the present invention.

[0037] 1.1 Introduction

[0038] Real-world communication channels are not stationary. Instead,they vary both in time and in frequency. Temporal fluctuations arecaused by moving objects in the environment. When waves originate from atransmitter that moves in relation to the receiver, or when waves bounceoff moving objects in the environment, they are shifted in frequency.These are the well-known “Doppler effects.” It should be noted that thespectrum of a transmitted waveform is unchanged, other than a frequencyshift, by the Doppler effect. The magnitudes of these effects depend onthe speed at which transmitters, receivers and objects are moving inrelation to each other, as well as the wavelength, λ, of the waves.These speeds change over time, thereby causing the temporal fluctuationsin the channels. In addition, received power changes over time becausewave attenuation changes: as objects move in the environment, reflectedwaves have to travel over varying distances from the transmitter to thereceiver.

[0039] Spectral fluctuations are caused by multiple echoes interferingat the point of reception. The way they interfere is determined by thepositions and scattering properties of the environmental features thatcaused the echoes (such as walls, furniture or people) relative to thewavelength. Since the wavelength, λ, is directly linked to thefrequency, f, by wave velocity, c=λf, different frequencies are treatedselectively by the channel, some interfering constructively, othersdestructively.

[0040] Channel fluctuations act on the transmitted waveforms by alteringtheir amplitudes, phases and spectral content. For the receiver, thesealterations obscure the information that was originally sent and thuspresent an obstacle in the process of correctly decoding the digitalsymbols. It is therefore crucial to reverse or compensate for the effectof the channel as much as possible by finding out what exactly thenature of the distortion is. This process of “learning” the channel isreferred to as channel estimation. Both temporal and spectral variationshave their share in the distortion, so in general both of them should betaken into account for an optimal estimation result and, ultimately,error free decoding.

[0041] 1.2 Modeling Temporal and Spectral Fluctuations

[0042] For the most general model for channels with both time andfrequency variation, the time domain is first examined. The time-variantimpulse response, h(t,τ), of a channel is given by the relation betweenthe transmitted waveform x(t) and the received waveform y(t) (It shouldbe noted that baseband notation is used here. The variables x, y, h, andtheir Fourier transforms are therefore in general complex valued):$\begin{matrix}{{y(t)} = {\int_{\tau = {- \infty}}^{0}{{h( {t,\tau} )}{x( {t - \tau} )}{{\tau}.}}}} & \text{(1.1)}\end{matrix}$

[0043] This model considers the channel as consisting of a continuum ofmulti-path components. For more information, see Proakis at page 764,the disclosure of which has already been incorporated by reference. Thevariable τ is the delay of each signal component x(t−r), and since τ iscontinuous, an infinite number of echoes is taken into account (referredto as “continuous scattering”). The attenuation, h(t,τ), of the echoesvaries over time τ due to Doppler effects, and it also varies with τ,due to the interference of echoes at the receiver. The received waveformis obtained by summing over all echoes with non-positive delays, sincephysical channels are causal systems.

[0044] The time-variant transfer function, H(t,f), is obtained by takingthe Fourier transform of the time-variant impulse response h(t,τ) asfollows: $\begin{matrix}{{H( {t,f} )} = {\int_{\tau = {- \infty}}^{+ \infty}{{h( {t,\tau} )}^{{- 2}\quad \pi \quad \quad f\quad \tau}{{\tau}.}}}} & \text{(1.2)}\end{matrix}$

[0045] If known, h(t,τ), or equivalently H(t,f), would completelycharacterize the channel. The main interest is in estimating either ofthese functions. The above expressions, which relate the input of thesystem to its output, cannot be simplified in the general case. As theglobal behavior of the real channel appears stochastic, the estimationproblem is approached by locally fitting the model function h(t,τ) tomeasured input and output data. For this purpose, the global signalingresources, time and bandwidth, are treated piecewise in conventionaltechniques.

[0046]FIG. 1 shows a known concept of dividing the time-frequency planeinto rectangles of duration, T, and bandwidth, B. These rectangles willbe referred to as time-frequency blocks 110, 120, 130, and 140. Block110, for example is centered at (t₁, f₁) and occupies an area equivalentto TB. Each block is associated with a burst, which is a packet ofdigital symbols modulated onto the physical waveforms that cover theblock. Most of the symbols will be the random data to be transmitted.Yet, in every block, some of the symbols have to be devoted to carryingknown values. These known values are the so-called “pilot” symbols, bywhich the unknown parameters of the channel model h(t,τ) are estimated.

[0047] The better the quality of these estimates, the fewer decodingerrors there are. If the model is not adequate, or if there are notenough pilot signals to obtain a good estimate, errors will occur. Inthat case, retransmission must occur and thus nothing has been gained bysaving on pilots. On the other hand, an arbitrarily accurate estimate ofthe channel cannot be an objective either, as this would require a largenumber of pilots. That strategy would not leave many resources fortransmitting data, the thing of primarily interest. Evidently, there isa fundamental trade-off between safely sending little data, or sending alot of data at a high risk of corruption. Given an upper limit ofdistortion (or decoding errors) that may be tolerated, the overallobjective is to minimize the required number of pilot symbols pertime-bandwidth area. This will be referred to as “pilot efficiency” or“estimation efficiency.”

[0048] For communication channels, better modeling must mean taking intoaccount both time and frequency variation, as both are actually present.Current technology focuses on circumventing the problem by choosing verysmall time-frequency blocks. What is lost through this technique ispilot efficiency. Poor modeling of the physical layer causes a good partof the limitations of today's wireless systems. In cellularcommunications, typically 10 percent of all symbols are pilots. Still,the number of symbol errors on the physical layer often is also on theorder of 10 percent or higher, even with this high pilot overhead.

[0049] While it is high in single-antenna wireless, the pilot overheadcan become overwhelming for MIMO systems. In these systems, multipletransmitters transmit over a channel to multiple receivers, which thenoutput decoded symbols from the multiple transmitters. This is describedin more detail in reference to FIG. 12. The problem with MIMO is thatthere are N² more coefficients to estimate than in the scalar case,where N is the number of transmit and also, generally, receive antennas.Since the propagation path from every transmit antenna to every receiveantenna is assumed to be independent from all other paths, there are N×Nindependent scalar time-varying transfer functions h(t,τ) to estimate.In general, all of them are time and frequency varying with the samecharacteristics as in the scalar case. The cost of “learning thechannel” thus grows with N², while even in the best case the totalnumber of symbols (or channel capacity) only grows linearly with N. Itis currently not clear how future MIMO systems would be able to carryany significant amount of data at all, if the current simple estimationstrategies were just to be carried over to the matrix case. The purenumbers show why, in practice, pilot efficiency becomes a criticalconsideration. High performance in terms of pilot efficiency is a keyfeature of the new kind of modeling principle proposed in the presentinvention.

[0050] 1.3 Three Simple Special Cases

[0051] As has been shown, the general channel model is the following:y(t) = ∫_(τ = −∞)⁰h(t, τ)x(t − τ)τ.

[0052] There are certain general preconditions that mark the limits asto which channels can be handled successfully by piecewise approximationusing blocks in the time-frequency plane. One could think ofmathematical functions for the behavior of the real channel that wouldmake communicating almost impossible, because the channel would beextremely incoherent. Basically, it is assumed that most of the energythat is transmitted within a certain time-bandwidth block does stay inthis block when passing through the channel. Channels are excluded thattransfer energy from the signaling band to any arbitrary otherfrequencies far out of the signaling band. Additionally, channels areexcluded that distribute significant amounts of energy over a very longtime scale. (Note that a long average delay or transit time betweensending and receiving waveforms does not affect the functioning of thescheme; instead, this is accounted for by synchronization.) Consideringthe physical causes of channel variation, it is plausible that those“bad” channels are not likely to occur. Time variation is caused byDoppler effects, which typically spread energy only to neighboringfrequencies, as moving objects usually do not move at arbitrarily highspeeds. Frequency variation is caused by echoes and these are notexpected to persist infinitely.

[0053] Within the mentioned limitations, one is free to choose a modelh(t,τ). Yet so far tools have been lacking to treat joint time-frequencyestimation within blocks. It is therefore common practice in wirelesscommunication systems to exploit three simpler special cases, that onlyinvolve either time or frequency variation. These are explained below.

[0054] 1.3.1 Time Translational Invariant Channel

[0055] The transfer function of a time translational invariant channelis modeled as follows:

h(t,τ)=h ₀(τ)

[0056] where h₀ is a fixed function regardless of time. In that case,input and output are directly related in the frequency domain (strictlyspeaking, this relation only holds for infinite window sizes):

Y(f)=H ^(c)(f)X(f)

[0057] where Y, H^(c) and X denote the Fourier transforms of y, h₀ andx, respectively. The time translational invariant model is appropriateif the actual temporal channel variation is very small compared to theduration of the signaling waveform, x(t). The Code-Division MultipleAccess (CDMA) modulation scheme assumes this to be the case (byengineering, i.e., choosing waveforms that are short enough). Theimpulse response can then be determined by sending broadband pilotsequences, waiting for all echoes to arrive back and auto-correlatingthe received waveform to determine the delays of the taps in the impulseresponse.

[0058]FIGS. 2A and 2B show the magnitude and phase, respectively, of atransfer function H(f) of an indoor acoustic channel between 1600 Hz and2000 Hz. This transfer function was measured in a cafeteria at a timewhen there was practically no movement. It shows the typicalcharacteristics also found for transfer functions of wireless channels:a large dynamic range with deep fades where the function drops by tendecibels or more, and relatively regular spacing of these fades.

[0059] A problem with this technique is it assumes that the actualtemporal channel variation is very small compared to the duration of thesignaling waveform. If this is not the case, then decoding errors canoccur.

[0060] 1.3.2 Stochastic Channel With “Flat Fading”

[0061] The transfer function of a stochastic channel with flat fading ismodeled as follows:

h(t,τ)=h ₁(t)δ(τ)

[0062] where δ, the Dirac impulse, means that all frequencies of theinput signal pass through the system without any delay. As aconsequence, there is no interference of echoes; the frequency spectrumis flat. The waveform is modulated over time by a stochastic amplitude,h₁(t). The output can simply be written as y(t)=h₁(t)x(t). In practice,one can engineer a system so that it works with this channel model, byconfining the signaling waveform to a narrow band. In physical systems,spectral variation is continuous. The channel is thus always reasonablyflat if one chooses the bandwidth interval small enough.

[0063]FIGS. 3A and 3B show magnitude and phase measurements,respectively, of time variation of the acoustic channel of a cafeteriaaround lunch time, when people were moving in the room and passing bythe experiment. The transmitted waveforms were two pure tones with timeinvariant amplitudes at 1500 Hz and 1520 Hz. Plotted are the receivedwaveforms (which are of course directly proportional to h₁(t) sincethere is no frequency dependence) over an interval of 18 seconds, afterdemodulation to the baseband and denoising. Clearly, the amplitudefluctuations in time are considerable, and this is only an indoorenvironment. Outdoors, the fluctuations could be much more rapid, forexample when cell phones are used in cars.

[0064] A problem with this technique is that the frequency spectrum isconsidered to be flat. If there are variations in frequency caused bythe channel, this approximation does not hold.

[0065] 1.3.3 Time Invariant and Flat Fading

[0066] The transfer function for a time invariant and flat fadingchannel is as follows:

h(t,τ)=h ₂δ(τ)

[0067] This simplest of all schemes assumes all frequencies pass equallywell at all times. The input waveform is only distorted bymultiplication with a complex constant transmission factor h₂, as itpasses through the channel. This channel model is still widely used incommunications, particularly in the Time Division Multiple Access (TDMA)and Orthogonal Frequency Division Multiplexing (OFDM) standards.Obviously both burst duration and bandwidth of the pulse shapingfunctions have to be chosen “small,” in order for the time invariant andflat fading assumptions to hold. If this is not true, then decodingerrors can occur.

[0068] 1.4 Channel Coherence and Modeling

[0069] In order to work well, each of the three models has differentrequirements in terms of channel coherence of the physical channel.Channel coherence is determined by coherence time and coherencebandwidth. For details about the definitions of coherence time andcoherence bandwidth see, for example, Proakis, which has beenincorporated by reference above. Basically, coherence time is large whenDoppler spread is small. Long coherence time means that the channel doesnot vary rapidly over time. Coherence bandwidth is large when the echoesdie down quickly at the receiver. Large coherence bandwidth means thatthere is not much variation over frequency. Consider FIG. 4A, avisualization of possible channel variation. Again, it should be notedthat the visualizations shown herein are not mathematically exact butare useful for description purposes. The warped hyperplane 405 shown inFIG. 4A represents the real part (the imaginary part could look similar)of a time and frequency varying channel, H(t,f), over one signalingblock, represented by the rectangle 410 in the time-frequency plane.

[0070] The time translational invariant channel approximates thishyperplane 405 with another hyperplane that is curved in the dimension(direction) of frequency, but flat in the dimension of time. See FIG.4A, where curve 420 is an example of what this other hyperplane mightlook like. As shown in FIG. 4A, curve 420 varies in frequency, but isthe same in time. In other words, at each “slice” of time, the curvelooks exactly the same. Obviously, the real channel H(t,f) is not flatin the dimension of time. Its coherence time is finite, but since thechannel works under the assumption of infinite coherence time, timevariation is not captured. The results are errors in channels estimationand thus Inter-Symbol Interference between the data symbols. The samething happens for the stochastic channel with “flat fading,” if onereplaces “time” with “frequency” in the above argument, and vice versa.

[0071] Even cruder is the estimation of the time invariant and flatfading channel: this zeroth-order scheme simply approximates thehyperplane 405 by a flat horizontal plane, with correspondingly largeerrors. This is illustrated in FIG. 4C, where horizontal plane 430 isused to estimate the hyperplane 405.

[0072] Given a certain model, a way to keep errors down is to choosesmaller signaling blocks. The problem is that, for every block, themodel coefficients must be estimated again. As one devotes resources(time and bandwidth) to pilot symbols, one wants to use them asefficiently as possible. A good model is a model that captures realityin an efficient way over wide ranges of the independent parameters t andf, meaning that it compresses a large part of the information containedin the channel variation in relatively few parameters. The minimumnumber of needed pilot symbols is proportional to the number of channelcoefficients to estimate. Good models therefore need few pilots. Sinceit is known that there are both temporal and spectral variation present,it is reasonable to expect that time-frequency models would be morepilot-efficient.

[0073] An example of a time-varying transfer function for atime-frequency model used in the present invention is shown in FIG. 4D.The time-varying transfer function of the present invention creates aplane 440 that can change in time and frequency and amplitude.Consequently, plane 440 models hyperplane 405 to a better degree than doother models.

[0074] 2 The First-Order Time-Frequency Model

[0075] This section introduces the first-order time frequency model, anddescribes the DPSS basis and the DPSS functions. It should be noted thatDPSS functions are commonly called “Slepian” functions. This name ischosen because of David Slepian, who was the first to find out thatthese functions were optimal solutions to the problem of simultaneoustime and frequency band limitation of a signal.

[0076] 2.1 Introducing Joint Time-Frequency Estimation

[0077] A scheme is presented herein that is capable of taking intoaccount both time and frequency variation. A first-order time-frequencychannel model is considered, making the following assumptions:

[0078] (1) The input time series, x(t), undergoes multiplication withlinear time and a complex-valued, constant factor H^(t) (the “time slopecoefficient,” since it is the first time derivative) as it passesthrough the channel. The time variation in the channel thus produces anoutput component of the following:

y ^((t))(t)=tH ^(t) x(t).

[0079] (2) The Fourier transform, X(f), of the input undergoesmultiplication with linear frequency and a complex-valued factor H^(f),the “frequency slope coefficient.” The frequency variation in thechannel thus produces an output component y^((f))(t), the Fouriertransform of which is given by the following:

Y ^((f))(f)=fH ^(f) X(f)

[0080] (3) Part of the input waveform passes through the channel withoutdistortion, and is only attenuated by a complex amplitude H^(c). Thecorresponding output component is y^((c))(t)=H^(c)x(t).

[0081] The complete received waveform is theny(t)=y^((c))(t)+y^((t))(t)+y^((f))(t). Unfortunately, this cannot beexpressed directly in terms of x(t), as y^((f))(t) is expressed in thefrequency (Fourier) basis, while y^((t))(t) is expressed in the timebasis. An important step in the approach taken herein is to use yetanother basis, one that deals naturally with both linear time andfrequency variation. Returning to the intuitive interpretation of FIG.4D, the model of the present invention approximates the hyperplane ofFIG. 4A by a plane segment 440 that is tilted in both directions, timeand frequency.

[0082] 2.2 The First-Order Channel Model

[0083] The channel model is a matrix model, the discrete time version ofEquation 1.1: $\begin{matrix}{y_{j} = {\sum\limits_{i = 1}^{n}{H_{ji}X_{i}}}} & \text{(2.1)}\end{matrix}$

[0084] Note that here i and j are discrete times, since time series aresampled in digital communications. The variable i takes the place of t−τin Equation 1.1. The variable n is the number of samples of the timeseries, and it is equal to the duration T of the time series, times thesampling rate 1/ΔT.

[0085] The choice of T must be guided by physical considerations. Onephysical consideration involves looking at measurements of channelstatistics like FIGS. 3A and 3B to determine a time interval over whichthe channel can be assumed to have only linear (first-order) variation.In a similar way, the signaling bandwidth, B, the “width” of thetime-frequency block, is chosen. Measurements of channel transferfunctions like FIGS. 2A and 2B indicate the coherence of the channel.The bandwidth, B, is chosen such that frequency variation typically isonly of no higher than first-order over intervals of that bandwidth.

[0086] For the following derivation of the channel model, indicialnotation (Einstein notation) is used that has the advantage of stayingreadable when there are a lot of indices involved. A short introductionto the Einstein notation is given in Appendix B of Karin Sigloch,“Communicating Over Multiple-Antenna Wireless Channels With BothTemporal and Spectral Fluctuations” (2002) (published Master's thesis,University of Karlsruhe, Germany), the disclosure of which is herebyincorporated by reference. In the following, the channel model for thesingle-antenna case is derived. The terms involved are therefore of nohigher than second order; expressions like “A_(lj)B_(ji)c_(i)” can thussimply be read as matrix and vector multiplications where i, j, and lare the indices into the matrices A and B and the vector c. This compactnotation is very useful when the channel model for the general MIMOcase, of N antennas on both transmitter and receiver side, is derived.The channel transfer function, H, is then a fourth-order tensor. Thefollowing indices are used: basis variable indices range time i, j 1, .. . , n frequency u, v 1, . . . , n Slepians k, l 1, . . . , K

[0087] The general channel Equation 2.1 in indicial notation is asfollows:

y _(j) H _(ji) x _(l)  (2.2)

[0088] The first-order time and frequency variations are modeled by thefollowing:

y _(j)=(H^(c)δ_(jl) +H ^(t) T _(jl) +H ^(f) F _(ljl))x _(l), and  (2.3)

y _(j)=δ_(jl) x _(l) H ^(c) +T _(jl) x _(l) H ^(f)  (2.4)

[0089] In matrix format, Equation 2.4 is shown as follows:$\begin{matrix}{y_{j} = \quad {\delta_{ji}\quad x_{i}\quad H^{c}\quad T_{ji}\quad x_{i}\quad H^{t}\quad F_{ji}\quad x_{i}\quad H^{f}}} \\{{\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix} = {{\begin{pmatrix}1 & \quad & \quad \\\quad & ⋰ & \quad \\\quad & \quad & 1\end{pmatrix}\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix}^{(\quad)}} + {\begin{pmatrix}{- \frac{T}{2}} & \quad & \quad \\\quad & ⋰ & \quad \\\quad & \quad & \frac{T}{2}\end{pmatrix}\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix}^{(\quad)}} + {\begin{pmatrix}\quad & \quad & \quad \\\quad & \quad & \quad \\\quad & \quad & \quad \\\quad & \quad & \quad\end{pmatrix}\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix}^{(\quad)}}}},}\end{matrix}$

[0090] where δ_(y) is the n x n identity matrix modeling the constantpart of the channel together with the complex gain H^(c). T_(jl) is then×n time operator and F_(ij) is the n×n frequency operator, both in thetime basis. In this “normal” basis for time series, the time operator issimply diagonal, with discrete time −T/2 . . . T/2 as its n diagonalentries.

[0091] To obtain the frequency operator in the time basis, the frequencybasis is first used, where the frequency operator is just diagonal withdiscrete frequency as its n diagonal entries. This is transformed intothe time basis by right and left multiplying it with the n-by-n discreteFourier transform matrix and its complex conjugate: $\begin{matrix}{F_{ij} =} & D_{iu} & F_{uv}^{(f)} & D_{vj} & \quad \\{\begin{pmatrix}\quad & \quad & \quad \\\quad & \quad & \quad \\\quad & \quad & \quad \\\quad & \quad & \quad\end{pmatrix} =} & \begin{pmatrix}\quad \\{DFT}^{\dagger} \\\quad\end{pmatrix} & \begin{pmatrix}{- \frac{B}{2}} & \quad & 0 \\\quad & ⋰ & \quad \\0 & \quad & \frac{B}{2}\end{pmatrix} & {\begin{pmatrix}\quad \\{DFT} \\\quad\end{pmatrix}.} & \quad\end{matrix}$

[0092]FIGS. 5A and 5B show plots of the time and frequency operators,respectively, in the time basis. The parameters chosen for these plotsare T=0.3 s, B=40 Hz, K=11, and n=75. What is plotted is representationsof the magnitudes of the complex matrix entries. In FIG. 5A, it can beseen that the time operator in the time basis is a diagonal matrix withnon-zero entries on the diagonal 510; all other entries are zero. FIG.5B shows the frequency operator, which is complex. The values for theareas are as follows: diagonal 520 contains zero values; area 530contains relatively high values; in area 540, there are values that arenot as high as the values in area 530, but are higher than zero; in area550, there are reasonably small values; the values in area 560 are closeto or equal to zero; in area 570, the values again increase; and area580 contains high values.

[0093] Thus, while the time operator looks very simple, the frequencyoperator has significant non-zero entries far off the diagonal. Thisconfirms that in the time basis, the relation between sent waveform x(t)and received waveform y(t) is complicated, involving all those non-zerocoefficients. Thus, an important step in the present invention is totransform the problem to another, more adequate basis. In this newbasis, time and frequency operators both look relatively simple, meaningthat they have clear structure and only few non-zero entries. This hasthe effect of limiting ISI.

[0094] 2.3 The Slepian Basis

[0095] A suitable basis is the Slepian basis. Its basis functions arethe DPSSs, called Slepian functions. Theoretical work on these functionsas the way to separate variables in the prolate spheroidal coordinatesystem goes back to about 1900. From the 1950's on, they were studied indepth by David Slepian in the context of the concentration problem inspectral analysis. See Slepian, “Prolate Spheroidal Wave Functions,Fourier Analysis and Uncertainty—I,” Bell Systems Technical Journal 40,43-63 (1961) and Slepian Prolate Spheroidal Wave Functions, “FourierAnalysis and Uncertainty—V: The Discrete Case,” Bell Systems TechnicalJournal 57, 1371-1430 (1978), the disclosure of which is herebyincorporated by reference.

[0096] The DPSSs are a set of orthonormal functions of length n points,where n=T/ΔT. More precisely, they are the set of orthonormal sequencesthat is maximally concentrated in energy within a chosen bandwidth B.FIGS. 6A through 6F show plots of the first six Slepian functions forT=2.4 s, B=5 Hz, K=11, and n=12000. One can compute n orthonormalSlepian functions, but, after the first K functions, there is a sharpdrop in the relative amount of energy concentrated in the band B. FIG. 7shows the concentration of energy in B over the number of the sequencefor the case T=2.4 s, B=5 Hz, K=11, and n=12000. For the communicationscheme presented herein, it is beneficial to use only the first KSlepians in order to avoid spilling of energy into neighboring bursts.For instance, as shown in FIG. 7, using K Slepians provides apredetermined energy concentration in band B of approximately 95percent. If the (K+1)-th Slepian is added for transmission of symbols,then the spillage into other bands is around 30 percent. Consequently,the number K is chosen to provide maximum Slepians (and thereforemaximum symbols that can be modulated) with a minimum of out-of-bandenergy spillage.

[0097] It is interesting to note that K=TB−1 is (almost) equal to thetime-bandwidth product TB of the block. The variable K is called thenumber of degrees of freedom of the time-frequency block. Looselyspeaking, K is the number of orthonormal functions that can beaccommodated by a time-frequency block of duration T and bandwidth B. Tounderstand the physical implications, recall the so-called uncertaintyrelation of time and frequency, named after Heisenberg's uncertaintyrelation of position and momentum in quantum mechanics. It states thatevery physical function has a time-bandwidth product of one or greater.An equivalent way of saying this is that every function must occupy atleast an area of one in the time-frequency plane. The time-frequencyblock in which the Slepians occupy has area TB. It would thereforeaccommodate at most TB non-overlapping functions, which, if used asprobing functions, would provide TB independent pieces of informationabout the channel. The same can be accomplished with the K (or K+1=TB ifone allows for some spilling) first Slepian functions. The difference isthat these functions are not distinguished by time slots or bandwidthintervals. Rather, they are independent because of their orthogonality,although each of them covers the whole T*B-rectangle. The number ofdegrees of freedom, K, where K=T*B−1, is therefore a measure of theindependent pieces of information that can be obtained about atime-frequency resource of area K. It does not matter in which way thefunctions are chosen not to interfere, one can theoretically never getmore than (roughly) TB of them.

[0098] 2.4 Mathematical Properties of the DPSSs

[0099] The DPSSs have some interesting properties. To start with, theyare their own Fourier transforms, except for a constant factor. Let{tilde over (ψ)}_(k)(f) be the Fourier transform of the k th Slepianψ_(k)(t). Then the relation is

ψ_(k)(t)=c{tilde over (ψ)}_(k)(t)  (2.5)

[0100] where c is a complex constant factor.

[0101] It is striking that the Slepian sequences look much like theeigenfunctions of the harmonic oscillator in quantum mechanics. In fact,both are solutions to similar concentration problems. Slepians arestrictly limited in one dimension, namely time, whereas the harmoniceigenfunctions are not. The eigenfunctions of the harmonic oscillatorare equivalent to the Hermite polynomials, except for normalization. TheHermite polynomials (and thus the eigenfunctions, except for constantfactors) are created by a simple recurrence relation:

H _(k+1)(x)=2xH _(k)(x)−2kH _(k−1)(x)

[0102] or

xH _(k)(x)=kH _(k−1)(x)+1/2H _(k+1)(x)  (2.6)

[0103] where x denotes position and H_(k) the k th Hermite polynomial.

[0104] It would be very interesting to know if the same kind ofrecurrence relation as Equation 2.6 holds for the Slepians, for thefollowing reason: the Slepians ψ_(k) would take the place of the H_(k),and time t the place of x. The question then is the following:

tψ _(k)(t)=c ₁ψ_(k−1)(t) Does this hold?  (2.7)

[0105] The right hand side multiplies the k th Slepian with a lineartime slope, exactly what is desirable to model. The left hand sidefeatures only the two neighboring Slepians of ψ_(k)—this means that thetime operator would be simply bi-diagonal in the Slepian basis. Evenbetter, the frequency operator would be bi-diagonal also: as Slepiansare their own Fourier transforms, Equation 2.7 would hold in thefrequency domain if Equation 2.5 is plugged in and the independentvariable t is renamed into f. Based on this argument, the followingwould hold, too:

f{tilde over (ψ)} _(k)(t)=c ₃{tilde over (ψ)}_(k+1)(t)+c ₄{tilde over(ψ)}_(k+1)(t)  (2.8)

[0106] It turns out that the Equations 2.7 and 2.8 do not hold exactly,since the Slepians are time-limited versions of the eigenfunctions. Italmost holds for small k, but spilling into higher functions, and thusout of the band, gets worse as k approaches K. Yet the followingrelation, which contains a projection back into the band, holds for thefirst K Slepians: $\begin{matrix}{{{\int_{- \infty}^{+ \infty}{{\psi_{k}(t)}t\quad {\psi_{l}(t)}{t}}} \cong {0\quad {unless}\quad {{k - 1}}}} = 1.} & (2.9)\end{matrix}$

[0107] The error of “≅” instead of “=” is small, the elements off thetwo bi-diagonals are orders of magnitude smaller than the elements onthe bi-diagonals. Thus the K×K time and frequency operators in the basisformed by the first K Slepians are practically bi-diagonal, while theywould not be if all N DPSSs were used. This is another reason forselecting the first K Slepians.

[0108] 2.5 Channel Model in the Slepian Basis

[0109] It has been shown that it is beneficial to choose the first KSlepians as basis functions (as they account for almost all of theenergy in the band B), and project the channel model from the n×n timebasis down to the K×K Slepian basis, where K is usually much smallerthan n. The following notation is used, shown in indicial notation:

[0110] (1) ψ_(ik) denotes the n×K matrix of the K Slepians.

[0111] (2) The Slepians are orthonormal:

ψ_(li)ψ_(ik)=δ_(lk)

[0112] (3) Modulation: let the transmitted waveform x_(i) be a linearcombination of the Slepians:

x _(l)=ψ_(lj) y _(j),

[0113] The {tilde over (x)}_(k) are the digital symbols (pilots and dataand guard symbol, if used) to be transmitted. Each of them is modulatedon a different Slepian. The scheme thus works with K pulse shapingfunctions.

[0114] (4) Demodulation: the received waveform y_(j) is demodulated withK matched filters:

[0115]  where {tilde over (y)}_(l) are the demodulated symbols.

[0116] The whole transmission equation in the Slepian basis is thus

{tilde over (y)} _(l)=ψ_(lj) y _(j)

{tilde over (y)} _(l)=104 _(lj)(H ^(c)δ_(ji) +H ^(t) T _(ji) +H ^(f) F_(ji))ψ_(ik) {tilde over (x)} _(k)

{tilde over (y)} _(l)=(ψ_(lj)δ_(ji)ψ_(ik) H ^(c)+ψ_(lj) T _(ji)ψ_(ik) H^(t)+ψ_(lj) F _(ji)ψ_(ik) H ^(f)){tilde over (x)} _(k)  (2.10)

{tilde over (y)} _(l)=(δ_(lk) H ^(c) +T _(lk) H ^(t) +F _(lk) H^(f)){tilde over (x)} _(k)

{tilde over (y)} _(l) =H _(lk) {tilde over (x)} _(k)  (2.11) In theSlepian basis, the matrix, vector, and scalar terms of Equation (2.10)have the following structure: $\begin{matrix}{\quad {{\overset{\sim}{y}}_{l} = \quad {\delta_{lk}\quad {\overset{\sim}{x}}_{k}\quad H^{c}\quad _{lk}\quad {\overset{\sim}{x}}_{k}\quad H^{t}\quad _{{lk}\quad}\quad {\overset{\sim}{x}}_{i}\quad H^{f}}}\quad} \\{{\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix} = {{\begin{pmatrix}1 & \quad & \quad \\\quad & ⋰ & \quad \\\quad & \quad & 1\end{pmatrix}\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix}^{(\quad)}} + {\begin{pmatrix}0 & ⋰ & 0 \\⋰ & 0 & ⋰ \\0 & ⋰ & 0\end{pmatrix}\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix}^{(\quad)}} + {\begin{pmatrix}0 & ⋰ & 0 \\⋰ & 0 & ⋰ \\0 & ⋰ & 0\end{pmatrix}{\begin{pmatrix}\vdots \\\vdots \\\vdots\end{pmatrix}^{(\quad)}\quad.}}}}\quad}\end{matrix}$

[0117] Equation 2.11 is a channel model that permits non-stationaritywithin one or more individual data symbols. Furthermore, Equation 2.11expresses a direct relation between the transmitted and receivedsymbols, not the whole waveforms as in the time basis. But as far as thecommunication problem goes, symbols are the only thing that anyone isreally interested in. So the problem is almost solved, as {tilde over(y)}_(l) are the received, distorted symbols, and {tilde over (x)}_(k)are the sent symbols, partly with known pilots, but mostly with unknowndata that is important to retrieve. The model coefficients H^(c), H^(t)and H^(f) are unknown and have to be estimated using the known pilotsymbols.

[0118] The variables T_(lk) and F_(lk) are the K×K time and frequency,respectively, operators in the Slepian basis, both (almost) bi-diagonal.FIGS. 8A and 8B show plots of the two operators in the Slepian basis forT=2.4 s, B=5 Hz, K=11, and n=12000. In FIG. 8A, the time operator isshown. The following values are associated with the time operator:elements 810 are zero; elements 820 are non-zero and relatively high invalue; elements 830 are non-zero and relatively low in value; andelements 840 are approximately zero. FIG. 8B shows the frequencyoperator for the Slepian basis. The following values are associated withthe frequency operator: elements 850 are zero; elements 860 are non-zeroand relatively high in value; elements 870 are non-zero and relativelylow in value; and elements 880 are approximately zero. Indeed, theelements off the two bi-diagonals are orders of magnitudes smaller thanthose on the bi-diagonals. The “constant” operator δ_(lk) is still theunity matrix, of size K×K in the Slepian basis.

[0119] The overall channel operator H_(lk) is the sum of the threebracketed terms in Equation 2.10, the sum of constant channel, timevariation and frequency variation. As T_(lk) and F_(lk) can beconsidered to be bi-diagonal and δ_(lk) is diagonal, H_(lk) hassignificant non-zero elements only on its diagonal and the twobi-diagonals. This has enormous implications on the decoding. When usingSlepians as pulse shaping functions, the value of the k th receivedsymbol {tilde over (y)}_(k) depends only on three sent symbols, namely{tilde over (x)}_(k−l), {tilde over (x)}_(k), and {tilde over(x)}_(k+l). Every symbol is distorted only by its two next neighbors,and in turn loses energy only to its two next neighbors. (This of courseis only true if the first-order time-frequency model is a goodapproximation of physical reality in the given TB block. But this can beshown to be true for a variety of situations.) This simple energyspilling pattern allows for quick decoding with a dynamic programmingalgorithm that will be described in section 3.1.5.

[0120] 2.6 Dealing With Synchronization Flaws

[0121] It is worth mentioning here that time-frequency estimation, eventhe relatively simple first-order scheme given here, compensates for twosynchronization flaws that commonly arise in digital communications:jitter in the timing of the synchronization instant, and drifts of thecarrier frequency. These are additional benefits to using aspects of thepresent invention.

[0122] 2.6.1 Errors in Timing

[0123] Assume that, due to imperfect clocks or uncertainty about thechannel, the moment at which a symbol starts in the received waveform isnot determined correctly. To be specific, assume that the receiverdetermines a symbol to start at time t₀+Δt where really it starts at t₀.The time delay or jitter Δt is interpreted by the demodulation procedureas a phase shift of 2πf₀Δt in the frequency domain, which is added tothe estimation H^(f) of the spectral fluctuation actually present in thechannel. The timing error thus hardly effects the accuracy of decodingat all.

[0124] 2.6.2 Drifts of the Carrier Frequency

[0125] Ideally, the carrier frequency should remain totally stable atf₀, but in practice some deviation is often encountered, which isrelatively slow, and small compared to f₀, but causes great distortionto the symbol after its demodulation to the baseband. Assume that thecarrier frequency actually is f₀+Δf during symbol demodulation. Thefrequency “delay” Δf is perceived by the estimation scheme as anadditional phase shift in time. It is accounted for by the estimation ofthe time slope H^(t), in complete analogy to the case discussed in theprevious section.

[0126] 3 Exemplary Apparatus and Implementation Issues

[0127] In this section, exemplary apparatus and actual implementation ofthe first-order time-frequency communication scheme are described.

[0128] 3.1 Elements of an Exemplary Communication Scheme

[0129] An exemplary communication between a receiver and a transmitterusing the techniques of the present invention is shown in FIG. 9.Transmitter 910 accepts input symbols 905 and outputs a transmit signal920. Generally, input symbols 905 are coupled to the transmitter 910 inblocks, as described above in reference to FIG. 1. The transmit signal920 is coupled to antenna 925 and broadcast over channel 930. Thetransmit signal 920 is received, after passing through channel 930 andexperiencing distortion caused by temporal and spectral effects of thechannel 930, by antenna 935 and becomes received signal 940. Thereceived signal 940 is coupled to the receiver 945, which performscertain operations to create decoded symbols 985. Ideally, decodedsymbols 985 are equivalent to input symbols 905.

[0130] Transmitter 910 comprises an DPSS modulator module 915, shown ingreater detail below. Broadly, DPSS modulator module 915 modulates eachinput symbol 905 with one DPSS function to create a modulated symbol,and then combines the modulated symbols to create transmit signal 920.

[0131] Receiver 945 comprises DPSS demodulator module 950, parameterdetermination module 970 and symbol decoding module 980. The DPSSdemodulator module 950 demodulates the received signal 940 by using thesame DPSS functions used by the DPSS modulator module 915. The DPSSdemodulator module 950 creates demodulated symbols 955, of which someare pilot symbols 960 and some are data symbols 965. The parameterdetermination module 970 uses the pilot symbols to determine theparameters 975 associated with Equation 2.10 (or Equation 2.11).Basically, the parameter determination module 970 is adapted todetermine parameters of a channel model that permits non-stationaritywithin individual data symbols. One exemplary technique for determiningthe parameters is discussed below. The parameters 975 are then used bysymbol decoding module 980 to create decoded symbols 985. This isdescribed in more detail below.

[0132] It should be noted that FIG. 9 represents a somewhat simplisticview of how a true communication system would appear. For instance,transmitter 910 will generally include additional digital modulationdevices (such as Binary Phase Shift Keying or Quadature AmplitudeModulation devices), frequency modulation devices to raise the transmitsignal 920 to a particular transmission frequency, and potentially errorcorrecting coding generation devices.

[0133] It should also be noted that DPSS modulator module 915, DPSSdemodulator module 950, parameter determination module 970, and symboldecoding module 980 can be implemented in software, hardware, or acombination of both.

[0134] 3.1.1 Modulation

[0135] The modulation procedure is summed up in FIG. 10 for the scalarcase. As can be seen in FIG. 10, there are K multipliers 1010-1 through1010-K (collectively, “1010”) for DPSS modulator module 915, eachmultiplier used to modulate one of the input symbols 905 with a DPSSfunction. The first six DPSS functions are shown in FIGS. 6A through 6F.Outputs of the multiplier are combined by combiner 1030 to createtransmit signal 920. For the DPSS demodulator module 950, there aremultipliers 1020-1 through 1020-K (collectively, “1020”), each of whichmultiplies the received signal 940 with a DPSS function to createdemodulated symbols 955, which are then shuttled to parameterdetermination module 970 and symbol decoding module 980 (not shown). Aspreviously discussed, the first six DPSS function are shown in FIGS. 6Athrough 6F.

[0136] In every burst, K symbols x_(i), i=1, . . . , K, are transmitted.Some of them, namely p, must be used to transmit pilot symbols known tothe receiver, to estimate the channel. The value of p must be chosen asp≧3, as there are three unknowns to estimate (H^(c),H^(t),H^(f)). Theremaining (K−p) symbols are the data to be transmitted, unknown to thereceiver.

[0137] 3.1.2 Pilot Placement

[0138] Each of the K symbols is modulated on one pulse shaping function(also called a DPSS symbol or Slepian). Although pilots may be placed onany of the K DPSS symbols, there are benefits to placing them onhigh-order DPSS symbols. In practice, the p pilot symbols on the last padjacent Slepians. The pilots are thus separated from the data block,which makes sure that pilot symbols spill only into other pilot symbols.As all pilots are known, this is a completely controlled and reversibleform of spilling.

[0139] A potential problem arises on the border of the pilot block andthe data block: the last data symbol spills into the first pilot symboland deteriorates the channel estimation. In practice, it is thereforechosen to modulate a zero on the (K−p) th Slepian. This is called a“guard” Slepian, by the means of which physical interaction between thedata and pilot blocks is eliminated. The guard Slepian is not necessarybut is beneficial. The pilots are carried by the higher order Slepiansbecause inter-symbol spilling is stronger among those than amonglower-order Slepians. For the estimation procedure this is a welcomeeffect, as it allows the time and frequency variations H^(t) and H^(f)to be estimated with greater precision. FIG. 11 shows a recommendeddistribution, in terms of Slepian symbols, of data and pilot symbols ina burst and how these symbols are ordered. As seen in FIG. 11, pilotplacement on the K Slepian pulse shaping functions occurs in a burst.The pilot block is separated from the data block by a gap, the “guardSlepian.” Keep in mind that this figure represents neither a sequentialorder in time nor in frequency, but in the Slepian basis. All symbolsare spread over the whole time-frequency block.

[0140] 3.1.3 Demodulation

[0141] All K waveforms {tilde over (x)}_(k)s_(k)(t) are added togetherand transmitted over the channel. The received signal y(t) isdemodulated by K matched filters which separate the K orthonormal pulseshaping functions. The resulting demodulated symbols are the {tilde over(y)}_(k).

[0142] 3.1.4 Estimation of the Three Channel Coefficients

[0143] Those {tilde over (x)}_(k) ^(p) that are pilot symbols, and thecorresponding {tilde over (y)}_(k) ^(p) are used to estimate the threechannel coefficients by minimizing $\begin{matrix}{{{{\overset{\sim}{y}}_{k}^{p} - {{\mathfrak{H}}_{kl}^{p}{\overset{\sim}{x}}_{l}^{p}}}},} & (3.1)\end{matrix}$

[0144] where H_(kl) ^(p) is the sub-matrix of the channel operatorH_(kl) that is associated with the pilot Slepians. If the following isplugged

H _(lk)=(δ_(lk) H ^(c) +T _(lk) H ^(t) +F _(lk) H ^(f))  (3.2)

[0145] in equation 3.1, it means that the parameters H^(c), H^(t) andH^(f) are fit to the measured data in a least mean squares sense.Equation 3.1 may be represented in a form in which it could be solvedwith a Matlab (an analysis program by The Mathworks, Natick, Mass.)solver or other software or hardware device, for any number of antennas,N. A program for performing the estimation scheme of the presentinvention has been performed on simulated and on real data. This isdescribed in more detail in chapters 4 and 5 of Karin Sigloch,“Communicating Over Multiple-Antenna Wireless Channels With BothTemporal and Spectral Fluctuations,” which has been incorporated byreference above.

[0146] Care has to be taken to achieve a good conditioning of thecorrelation matrix that has to be inverted in the course of the leastmean squares estimation. This matrix can be engineered in the sense thatthe choice of good pilots is up to the designer. It is possible toderive an analytical form for the best pilots. It is also possible touse Monte Carlo optimization to find good condition numbers for thematrix. In general, better condition numbers can be obtained if one isprepared to increase the number of pilots. For the results presented inchapter 5 of Sigloch, the author worked with condition numbers around 4,and chose p=4 or 5. Once the estimate H_(lk) of the channel is known,and one can proceed to the decoding of the data symbols.

[0147]3.1.5 Quick Decoding Algorithm

[0148] As described previously, every demodulated symbol {tilde over(y)}_(k) depends only on three transmitted symbols. For the kth receivedsymbol, all possible triplets ({tilde over (x)}_(k−1), {tilde over(x)}_(k), {tilde over (x)}_(k+1)) have to be considered as potentiallytransmitted sequences, but this is much less than considering allcombinations of K symbols. Moreover, the Slepians are linked together inthe sense that {tilde over (x)}_(k−1), {tilde over (x)}_(k), {tilde over(x)}_(k+1) affect {tilde over (y)}_(k), but {tilde over (x)}_(k) and{tilde over (x)}_(k+1) also affect the next received symbol {tilde over(y)}_(k+1). So even if one combination of ({tilde over (x)}_(k−1),{tilde over (x)}_(k), {tilde over (x)}_(k+1)) seems to explain well thereception of {tilde over (y)}_(k), there will still be a penalty if itdoes not explain well the reception of {tilde over (y)}^(k−1) and {tildeover (y)}_(k+1). Finding the best fitting sequence {tilde over (x)}₁ . .. , {tilde over (x)}_(K) is a dynamic programming problem. Its depth isthree, as three neighboring signals contribute to each {tilde over(y)}k. This kind of problem is well known in communications, as well asways of tackling it. Viterbi's algorithm is usually used as a quick andconvergent method of decoding and may be used here. Those skilled in theart can design and implement an adapted version of this algorithm forany number of antennas.

[0149] 3.1.6 Extension to the Matrix Case

[0150] Multiple-In-Multiple-Out (short MIMO) systems are a recentdevelopment in wireless communications where multiple antennas are usedboth on the transmitting and receiving sides. For the same totaltransmit power, signaling bandwidth, and signaling duration, MIMOsystems can achieve much higher channel capacities than systems withonly one antenna on each side. In the best case, that is if there is thesame number N of antennas at the transmitter and the receiver, and ifthe channel is flat-fading with Gaussian noise, the theoretical channelcapacity at high signal-to-noise ratios is N times higher than for ascalar system under the same conditions, using the same signalingresources.

[0151] The principle of MIMO communications is that in addition totemporal and spectral diversity, another form of diversity is exploitedto distinguish data streams originating from different antennas. Spatialdiversity, and recently polarization diversity, have been proposed andshown to work in experiments. Now, a brief discussion of how MIMO workswill be given. Let M be the number of antennas at the transmitter, and Nthe number of antennas at the receiver. The M transmit antenna send outM independent data streams, all of them simultaneously and over the samebandwidth. The M signals get scattered by the environment, and each ofthe N receive antennas receives a superposition of waves originatingfrom all M transmit antennas. The continuous, time-variant transferfunction h(t,τ) (Equation 1.1) is then a matrix of dimension N×M, sincethe paths from every transmit antenna to every receive antenna all havedifferent transfer coefficients. The crucial assumption is that theenvironment provides rich scattering. The central paradigm shift, whengoing from current scalar techniques to MIMO communications, is that inMIMO scattering is treated as an ally rather than an adversary. If inspatial MIMO transmit as well as receive antennas are sufficientlyspaced from each other (about one wavelength or more for diffuseenvironments), then all M data streams are scattered independently. Inthat case, the transfer matrix has full rank, and the sent signals canbe retrieved from the received signals by inverting the channel matrix.In polarization MIMO, all M data streams originate from the same spot inspace, but they are modulated onto differently polarized waves. M is atmost six, as there are six degrees of freedom in polarization, threeorthogonal directions of the electric and three of the magnetic field.The polarized components are scattered independently, depending on therandom angles at which the waves hit the scatterers. The data streamsmodulated onto differently polarized carriers can thus be separated atthe receiver.

[0152] The current wave of MIMO research started with seminalcontributions on the theoretical capacity limits of multiple-antennachannels in scattering environments, by Foschini, Gans and Teletar. SeeFoschini, “Layered Space-Time Architecture for Wireless Communication ina Fading Environment When Using Multiple Antennas,” Bell Labs TechnicalJournal, Vol. 1, No. 2, 41-59 (Autumn 1996); Foschini et al., “On Limitsof Wireless Communications in a Fading Environment When Using MultipleAntennas,” Wireless Personal Communications, Vol. 6, No. 3, 311 (March1998); and Teletar, “Capacity of Multi-Antenna Gaussian Channels,”European Transactions on Telecommunications, (1999), the disclosures ofwhich are hereby incorporated by reference. Another importantcontribution was the publication of the BLAST algorithm, acomputationally efficient algorithm for the detection of the M signalsat the receiver. For more information about the BLAST algorithm, seeGolden et al., “Detection Algorithm and Initial Results Using theV-BLAST Space-Time Communication Architecture,” Electronics Letters,Vol. 35, No. 1, 14-15 (January 1999), the disclosure of which is herebyincorporated by reference. The computational cost of MIMO isconsiderable compared to scalar systems. Polarization MIMO was proposedand demonstrated in experiments by Andrews, Mitra and deCarvalho in2001. See Andrews et al., “Tripling the Capacity of WirelessCommunications Using Electromagnetic Polarization,” Nature 409, 316-318,(January 2001), the disclosure of which is hereby incorporated byreference.

[0153] An even bigger hurdle than channel inversion may be the channelestimation problem. The derivations of capacity limits assume thechannel to be known. In practice, knowing the channel means estimatingit often enough, and the number of pilots needed to learn it can be solarge that not much resources remain for data. The basic problem, asexplained more in detail in Section 1 above, is that the even in thebest case, with N antennas on either side, the total number of symbolsin MIMO is only N times what it would be in the single antenna case.Unfortunately, the number of pilot symbols needed to estimate the N×NMIMO channel matrix grows much faster than that, namely with N².

[0154] The extension to the multiple antenna case is conceptuallystraightforward, although sometimes tricky in practical implementation.Mathematics for the matrix case are worked out below. A diagram of aMIMO communication is shown in FIG. 12, where two transmitters(comprising DPSS modulator modules 915-1 and 915-2) transmit using twoantennas 925-1 and 915-2 over channel 930 to antennas 935-1 and 935-2 oftwo receivers (comprising DPSS demodulator modules 950-1 and 950-2).DPSS demodulator module 950-1 basically interacts with two differentchannel operators, h₁₁ and h₁₂. Similarly, DPSS demodulator module 950-2interacts with the channel operators h₂₁, and h₂₂. The parameters H^(c),H^(t) and H^(f) are each 2×2 matrices and the demodulated symbols 955-1and 955-2 are used to determine these parameters. A recommended schemefor modulating data and pilot symbols on Slepian functions is shown inFIG. 13.

[0155] The system of FIG. 12 may be extended to N transmitters and Nreceivers. Each of the N transmit antennas transmits one of the burstsdescribed above, except that one now has to use pN² Slepians for pilots,where p≧3, because the channels H^(c), H^(t) and H^(f) are N×N matrices,each with N² complex coefficients to estimate.

[0156] The N transmit antennas each send a waveform x_(α)(t), α=1, . . ., N, and each of the N receive antennas receives a waveform y_(β)(t),β=1, . . . , N. Every y_(β)(t) is a superposition of contributions fromall N transmit antennas, i.e., distorted versions of x_(α)(t). Thedemodulation and decoding works as in the scalar case, except that nowthe dynamic decoding algorithm has to treat vectors of symbols insteadof single symbols.

[0157] Appendix: Mathematics for MIMO

[0158] The following indices will be used: basis variable indices rangetime i, j 1, . . . , n frequency u, v 1, . . . , n Slepians k, l 1, . .. , K antennas α, β 1, . . . , N

[0159] As shown above, the transmission equation for the scalar channelis as follows:

y _(j) =H _(ji) x _(i).

[0160] In addition to describing the transition from relative time i toabsolute time j, the MIMO channel equation will also have to express thewave propagation from every transmit antenna to every receive antenna.The variable α is introduced as transmit antenna index and the variableβ is introduced as receive antenna index. The most general channelequation for the multi-antenna case is then

y _(jβ) =H _(jiβα) x _(iα)  (A.1)

[0161] The first-order time and frequency variations are modeled by$\begin{matrix}{y_{j\beta} = {( {{H_{\beta\alpha}^{c}\delta_{ji}} + {H_{\beta\alpha}^{t}T_{ji}} + {H_{\beta\alpha}^{f}F_{ji}}} )x_{i\alpha}}} & ( {A{.2}} ) \\{y_{j\beta} = {{\delta_{ji}x_{i\alpha}H_{\alpha\beta}^{c}} + {T_{ji}x_{i\alpha}H_{\alpha\beta}^{t}} + {F_{ji}x_{i\alpha}H_{\alpha\beta}^{f}}}} & ( {A{.3}} )\end{matrix}$

[0162] Note that the time and frequency operators are the same as in thescalar case. The channel coefficients H_(αβ)^(c), H_(αβ,)^(t)

[0163] and H_(αβ)^(f)

[0164] are now N×N matrices, that have to be estimated. Thetransformation to the Slepian basis works in the same way as for thescalar case:

[0165] (1) ψ_(ik) denotes the n×K matrix of the K Slepians.

[0166] (2) The Slepians are orthonormal:

ψ_(li)ψ_(ik)=δ_(lk)

[0167] (3) Modulation: let the transmitted waveform at antenna α,x_(iα),be a linear combination of the Slepians:

x _(iα)=ψ_(ik) {tilde over (x)} _(kα),

[0168]  where the {tilde over (x)}_(kα) are the digital symbols (pilotsand data) to be transmitted by antenna α. Each of them is modulated ontoa different Slepian. The scheme thus works with K pulse shapingfunctions.

[0169] (4) Demodulation: the received waveform y_(jβ) is demodulated byK matched filters:

{tilde over (y)} _(lβ)=ψ_(lj) y _(jβ),

[0170]  where {tilde over (y)}_(lβ) are the demodulated symbols atantenna β.

[0171] The whole transmission equation in the Slepian basis is$\begin{matrix}{{{{\overset{\sim}{y}}_{l\beta} = {\Psi_{lj}y_{j\beta}}}{{\overset{\sim}{y}}_{l\beta} = {{\Psi_{lj}( {{\delta_{ji}H_{\beta\alpha}^{c}} + {T_{ji}H_{\beta \quad \alpha}^{t}} + {F_{ji}H_{\beta\alpha}^{f}}} )}\Psi_{ik}{\overset{\sim}{x}}_{k\alpha}}}{{\overset{\sim}{y}}_{l\beta} = {( {{\Psi_{lj}\delta_{ji}\Psi_{ik}H_{\beta\alpha}^{c}} + {\Psi_{lj}T_{ji}\Psi_{ik}H_{\beta \quad \alpha}^{t}} + {\Psi_{lj}F_{ji}\Psi_{ik}H_{\beta\alpha}^{f}}} ){\overset{\sim}{x}}_{k\alpha}}}{{\overset{\sim}{y}}_{l\beta} = {( {{\delta_{ik}H_{\beta\alpha}^{c}} + {_{ik}H_{\beta\alpha}^{t}} + {\mathcal{F}_{lk}H_{\beta\alpha}^{f}}} ){\overset{\sim}{x}}_{k\alpha}}}}} & ( {A{.4}} ) \\{{\overset{\sim}{y}}_{l\beta} = {\mathcal{H}_{lk}{\overset{\sim}{x}}_{k\alpha}}} & ( {A{.5}} )\end{matrix}$

[0172] The last formula directly relates the symbols {tilde over(x)}_(kα) transmitted by antenna α, to the symbols {tilde over (y)}_(lβ)received by antenna β. The decoding works as for the scalar case, onlythat now it is necessary to compare vectors of N symbols. Lety_(β)^((k))

[0173] be the N-dimensional vector of symbols that consists of all Nsymbols demodulated from Slepian number k. This vector will then dependon x_(α)^((k − 1)), x_(α)^((k))  and  x_(α)^((k + 1)),

[0174] that is, all 3N symbols that were modulated onto Slepians numberk−1, k and k+1. This ends the mathematics of the MIMO communicationsituation.

[0175] 4. Conclusion

[0176] The scheme described is the first known attempt to do truly jointtime-frequency channel estimation in wireless communications. As thismodel is closer to physical reality than the current zeroth-orderapproach, it is expected that it will turn out to be considerably morepilot-efficient in practice. Moreover the first-order time-frequencyscheme offers a computationally elegant way of decoding, due to verycontrolled energy spilling within and across bursts in the Slepianbasis. A nice by-product of time-frequency estimation is the robustnesstowards synchronization errors.

[0177] It is to be understood that the embodiments and variations shownand described herein are merely illustrative of the principles of thisinvention and that various modifications may be implemented by thoseskilled in the art without departing from the scope and spirit of theinvention.

We claim:
 1. A method used in communication systems, comprising:determining one or more parameters of a channel model that permitsnon-stationarity within one or more individual data symbols; andapplying the one or more parameters to one or more received symbols todetermine one or more decoded symbols.
 2. The method of claim 1, whereinthe channel model is modeled as a transfer function relating transmittedsymbols to received symbols, the transfer function comprising the one ormore parameters.
 3. The method of claim 1, wherein the parameters of thechannel model comprise a time slope coefficient, a frequency slopecoefficient, and a complex amplitude.
 4. The method of claim 1, furthercomprising the step of demodulating a received signal by using one ormore Discrete Prolate Spheroidal Sequences (DPSSs) to create thereceived symbols.
 5. The method of claim 4, further comprising the stepof demodulating the received signal by using a plurality of DPSSs, thestep of demodulating producing a received symbol for each of theplurality of DPSSs.
 6. The method of claim 5, wherein a firstpredetermined number of the received symbols correspond to known pilotsymbols, and wherein the step of determining the one or more parametersfurther comprises solving a system of equations using the known pilotsymbols to determine the parameters.
 7. The method of claim 6, whereinthe system of equations is represented through an equation having aplurality of matrices, the equation comprising a reception matrix havingreceived symbols corresponding to the known pilot symbols, a pilotmatrix having the known pilot symbols, and a channel model matrix havingthe parameters, wherein the equation is the reception matrix minus aproduct of the channel model matrix and the pilot matrix.
 8. The methodof claim 7, wherein the step of determining one or more parametersfurther comprises the step of minimizing a magnitude of the equation. 9.The method of claim 7, wherein a second predetermined number of thereceived symbols correspond to data symbols, and wherein the step ofapplying further comprises the step of applying a second system ofequations to the data symbols, the second system of equations comprisingthe parameters.
 10. The method of claim 9, wherein: the parameters ofthe channel model comprise a time slope coefficient, a frequency slopecoefficient, and a complex amplitude; the data symbols are representedby a data vector; and the second system of equations comprises: a firstmultiplication of a unity matrix, the data vector, and the complexamplitude; a second multiplication of a time operator matrix, the datavector, and the time slope coefficient; a third multiplication of afrequency operator matrix, the data vector, and the frequency slopecoefficient; and an addition of results of the first, second, and thirdmultiplications, whereby the decoded symbols result from the addition.11. The method of claim 3, wherein there are a plurality of antennasreceiving signals and wherein the time slope coefficient, frequencyslope coefficient, and complex amplitude are matrices with a number ofentries corresponding to a square of the number of antennas, wherein themethod further comprises the step of demodulating, for each of thereceivers, a respective one of the signals to create the receivedsymbols, and wherein the step of determining further comprises the stepof determining the parameters by using pilot symbols in received symbolsfrom each of the receivers.
 12. An apparatus used in a communicationsystem, comprising: a parameter determination module adapted todetermine one or more parameters of a channel model that permitsnon-stationarity within one or more individual data symbols; and asymbol decoding module coupled to the parameter determination module andadapted to apply the one or more parameters to one or more receivedsymbols to determine one or more decoded symbols.
 13. The apparatus ofclaim 12, wherein the channel model is modeled as a transfer functionrelating transmitted symbols to received symbols, the transfer functioncomprising the one or more parameters.
 14. The apparatus of claim 12,wherein the parameters of the channel model comprise a time slopecoefficient, a frequency slope coefficient, and a complex amplitude. 15.The apparatus of claim 12, further comprising a demodulator modulecoupled to both the parameter determination module and the symboldecoding module, the demodulator module demodulating a received signalby using one or more Discrete Prolate Spheroidal Sequences (DPSSs) tocreate the received symbols.
 16. The apparatus of claim 15, wherein: thedemodulator is further adapted to demodulate the received signal byusing a plurality of DPSSs, the demodulating producing a received symbolfor each of the plurality of DPSSs; a first predetermined number of thereceived symbols correspond to known pilot symbols and the apparatus isadapted to couple the pilot symbols to the parameter determinationmodule; a second predetermined number of the received symbols correspondto data symbols and the apparatus is adapted to couple the data symbolsto the symbol decoding module; and the parameter determination module isfurther adapted to solve a system of equations using the known pilotsymbols to determine the parameters.
 17. A method used in acommunication system, comprising: modulating one or more symbols withone or more Discrete Prolate Spheroidal Sequences (DPSSs) to createmodulated symbols; and combining the modulated symbols.
 18. The methodof claim 17, further comprising the step of transmitting the modulatedsymbols over one or more antennas.
 19. The method of claim 17, whereinthe step of modulating creates a signal and wherein the step ofmodulating further comprises the step of modulating each of a pluralityof symbols with one of a plurality of distinct DPSSs, whereby the signalhas a predetermined time duration and bandwidth.
 20. The method of claim19, wherein the step of modulating further comprises the step ofmodulating a predetermined number of data symbols with firstpredetermined ones of the distinct DPSSs.
 21. The method of claim 20,wherein the step of modulating further comprises the step of modulatinga predetermined number of pilot symbols with second predetermined onesof the distinct DPSSs.
 22. The method of claim 21, wherein thepredetermined ones of the distinct DPSSs are the highest-ordered DPSSsused to modulate the data and pilot symbols.
 23. The method of claim 21,wherein the predetermined ones of the distinct DPSSs are chosen to becontiguous in terms of order, wherein the step of modulating furthercomprising separating the plurality of data symbols by modulating datasymbols with a first predetermined number of contiguous, as determinedby order, distinct DPSSs and by modulating a guard symbol for one of thedistinct DPSSs, the DPSS for the guard symbol chosen as having an orderbetween the orders of the DPSSs used to modulate the data symbols andthe orders of the DPSSs used to modulate the pilot symbols.
 24. Themethod of claim 19, wherein a number of the distinct DPSSs used tomodulate both the data symbols and the pilot symbols is chosen so that apredetermined energy concentration for all of the DPSSs exists in abandwidth of the signal.
 25. The method of claim 24, wherein the numberof the distinct DPSSs is chosen to be the time duration multiplied bythe bandwidth minus one, whereby the predetermined energy concentrationis approximately 95 percent.
 26. An apparatus used in a communicationsystem, comprising: a plurality of modulators, each modulator adapted tomodulate a symbol with a Discrete Prolate Spheroidal Sequence (DPSS);and a combiner coupled to the modulators and adapted to combine resultsof the plurality of modulators to create a signal.
 27. The apparatus ofclaim 26, further comprising one or more antennas coupled to the signaland adapted to transmit the signal.
 28. The apparatus of claim 26,wherein each of the modulators is adapted to modulate one of a pluralityof symbols with one of a plurality of distinct DPSSs, whereby the signalhas a predetermined time duration and bandwidth.
 29. The apparatus ofclaim 28, wherein a first predetermined number of the modulatorsmodulate data symbols, and wherein a second predetermined number ofmodulators modulate pilot symbols, wherein the distinct DPSSs used tomodulate the pilot symbols are highest-ordered DPSSs used to modulatethe data and pilot symbols.
 30. The apparatus of claim 29, wherein anumber of the distinct DPSSs used to modulate both the data symbols andthe pilot symbols is chosen so that a predetermined energy concentrationfor all of the DPSSs exists in a bandwidth of the signal.